CERTAIN RESULTS ON $\eta$-RICCI SOLITIONS AND ALMOST $\eta$-RICCI SOLITONS

Abstract

We prove that if an $\eta$-Einstein para-Kenmotsu manifold admits a $\eta$-Ricci soliton then it is Einstein. Next, we proved that a para-Kenmotsu metric as a $\eta$-Ricci soliton is Einstein if its potential vector field $V$ is infinitesimal paracontact transformation or collinear with the Reeb vector field. Further, we prove that if a para-Kenmotsu manifold admits a gradient almost $\eta$-Ricci soliton and the Reeb vector field leaves the scalar curvature invariant then it is Einstein. We also construct an example of para-Kenmotsu manifold that admits $\eta$-Ricci soliton and satisfy our results. We also have studied $\eta$-Ricci soliton in 3-dimensional normal almost paracontact metric manifolds and we show that if in a 3-dimensional normal almost paracontact metric manifold with $\alpha, \beta $ = constant, the metric is $\eta$-Ricci soliton, where potential vector field $V$ is collinear with the characteristic vector field $\xi$, then the manifold is $\eta$-Einstein manifold.

A study on conformal Ricci solitons and conformal Ricci almost solitons within the framework of almost contact geometry

“…By virtue of ( 7), the preceeding equation reduces to X( f + λ) = ξ( f + λ)η(X) for any X ∈ χ(M). We can conclude from Lemma 3 that λ + f = c is a constant on M. Contracting (10) provides…”